Ўзбекистон миллий университети ҳузуридаги илмий даражалар берувчи dsc. 03/30. 12. 2019. Fm. 01. 01 Рақамли илмий кенгаш математика институти
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Автореферат Ботиров
- Bu sahifa navigatsiya:
- “Gibbs measures for a model with the set [0,1] of spin values
- Proposition 5.
- Proposition 7.
- Proposition 9.
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Theorem 1. A sequence probability distributions 1,2, = ), ( ) (
n n σ µ given by (2) are compatible iff for any 0 \ { } x V x ∈
(
( )
y u f u h u J du y u f u h u t J x t f x S y ) , ( ) ( cos ) ( cos exp
) , ( ) ( cos ) ( cos exp = ) , ( 2 0 2 0 ) ( β β β β π π − − − ∫ ∫ ∏ ∈ (4)
Here ) [0,2 ), ( exp = ) , ( 0, , π ∈ − t h h x t f x x t and
) ( = du du λ is the Lebesgue measure. Theorem 2. Let for the temperature the following inequality holds: k k h J k T 2 4 1 1 ln |) | | (| 2 > 2 + + + (5) Then the model (1) has a unique Gibbs measure. In the second chapter, titled “Gibbs measures for a model with the set [0,1] of spin values” several model with competing interactions and a continuum spin values are constructed, which have at least two periodic Gibbs measures. In the first section of Chapter 2, we study the phase-transition behavior of nearest-neighbor model on Cayley trees with arbitrary degree 2 ≥
. For
Φ = [0,1] we consider the following Hamiltonian on Ω: ) , ( ) ( , ) ( ) ( , β θ ξ σ σ σ β θ ∑ >∈ < − = = L y x y x J H H , (6) where 𝐽𝐽 ∈ ℝ ∖ {0} and 𝜉𝜉: (𝑢𝑢, 𝑣𝑣) ∈ [0,1] 2 → 𝜉𝜉
𝑢𝑢,𝑣𝑣 ∈ ℝ is a given bounded, measurable function.
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For every 𝑘𝑘 ∈ ℕ we consider an integral operator k H acting in the cone ] 1 , 0 [ + C as follows ( )( ) ∫
∈ = 1 0 , ) ( ) , ( N k du u f u t K t f H k k .
(7)
The operator k H is called Hammerstein’s integral operator of order k . We
notice that if 2 ≥ k then
k H is a nonlinear operator. From works of U.Rozikov, Yu.Eshkabilov and F.Haydarov it is known that the set of translational invariant Gibbs measures of the model (6) is described by the fixed points of the Hammerstein’s operator such that ( ) ( )
1 0 1 ( ) 0 ( , ) , ( , )
(0, ) ,
K t u f u y du f t x K u f u y du ∈ = ∫ ∏ ∫ ,
(8)
where ( ) ( , ) exp
tu K t u J βξ = , and ( )
du du λ = Lebesgue measure.
Let 2 ≥
and we suppose that function ( ) ( )
x y σ σ ξ in the model (6) is ( )
1 , 0 , , 2 1 2 1 4 1 ln 1 , 1 2 , , ∈ − − + = = + u t u t n u t u t θ β β θ ξ ξ
(9) where 1 0 < ≤ θ . Then for the kernel ) , ( u t K of the Hammerstein’s operator k H we
have ( )
1 2 2 1 2 1 4 1 , + − − + = n u t u t K θ . (10)
Here we use the notation
− = odd is if , 1 even
is if , | | even k k k k k and − = even. is if , 1 odd is if , | | odd k k k k k
𝑉𝑉 𝑘𝑘 : (𝑥𝑥, 𝑦𝑦) ∈ ℝ 2 → (𝐶𝐶
1 , 𝐶𝐶
2 ) ∈ ℝ
2 by
( ) ( )
⋅ + + + ⋅ + + + = ∑ ∑ = − + − = − + odd even
| | ,..., 3 , 1 1 2 1 | | ,..., 2 , 0 1 2 , 2 2 2 1 2 2 1 2 1 2 ) , ( k i i i k n i k i i i k n i n k y x i n n i k y x i n n i k y x V θ θ
(11) Proposition 3. A function ] 1 , 0 [ C ∈ ϕ is a fixed point of Hammerstein’s integral operator equation of order k iff ) (t ϕ has the following form 37
, ) 2 1 ( 4 ) ( 1 2 2 1 + − + = n t C C t θ ϕ where (𝐶𝐶
1 , 𝐶𝐶
2 ) ∈ ℝ
2 is a fixed point of the operator n k V ,
𝑛𝑛 ∈ ℤ
+ and 2 ≥ k . If ) 1 2 ( 3 2 + + = n k n c θ
following statements hold: (i) there exists a unique translation-invariant splitting Gibbs measure if c θ θ ≤ ≤ 0 ; (ii) there exist exactly three translation-invariant splitting Gibbs measures if 1
θ θ
In the second section of Chapter 2, we consider a phase transition for the model at k=2 and k=3. For 3 =
and n=1 the operator 𝑉𝑉 3,1 : (𝑥𝑥, 𝑦𝑦) ∈ ℝ 2 → (𝑥𝑥 ′ , 𝑦𝑦
′ ) ∈ ℝ
2 has the form
⋅ + = ⋅ + = 3 3 3 2 2 3 2 3 2 7 6 5 9 ' , 2 5 18 ' y y x y xy x x θ θ θ
(12) Proposition 5. а) If 9 5 0 ≤ ≤ θ , then Hammerstein operator of order three has a unique nontrivial positive fixed point; b) If 1 9 5 < < θ
operator of order three. Consequently, by Proposition 5 the operator H 3 has a unique positive fixed point ( )
1 ) ( 1 ≡ = t t ϕ ϕ if 9 5 0 ≤ ≤ θ . In the case 1 9
< < θ the functions 1 ) ( 1 ≡
ϕ , 3 1 1 2 2 1 4 ) ( − + = + + t y x t θ ϕ , 3 1 1 2 2 1 4 ) ( − + = − +
y x t θ ϕ , are positive fixed points of the Hammerstein’s operator H 3 .
following statements hold: а) If 9 5 0 ≤ ≤ θ , then there exists a unique translational-invariant Gibbs measure; b) If 1 9 5 < < θ
measures. In the third section of Chapter 2, we consider the model with a bifurcation analysis for any , 2 ≥ k case
1 =
and we show that for
3 5 0 ≤ ≤ θ the model has a 38
unique translation-invariant Gibbs measure, and for 1 3 5 < < θ
there is a phase transition, in particular there are three translation-invariant Gibbs measures. Let 2
k and n=1. The function V k,1 can be written in the following way. First for even 2 ≥
: For even 2 ≥
For odd
3 ≥
( )
( ) ( )
( ) = = = ∑ ∑ − = − = − 1 ,..., 3 , 1 3 ,...,
2 , 0 3 2 3 ' 2 3 ' ) , ( k l k l k l k l k l k l k l B y x l k y l A y x l k x y x V θ θ ( )
( ) ( )
( ) = = = ∑ ∑ − = − = − 1 ,..., 2 , 0 3 ,...,
3 , 1 3 2 3 ' 2 3 ' ) , ( k l k l k l k l k l k l k l B y x l k y l A y x l k x y x V θ θ Proposition 7. а) If k 3 5 0 ≤ ≤ θ , then the Hammerstein operator of order k has a unique (nontrivial) positive fixed point; b) If 1 3 5 < < θ
, then the Hammerstein operator of order k has exactly three positive fixed points. Consequently, by Proposition 7, the operator H k has a unique positive fixed point ( )
1 1 ≡ t ϕ if k 3 5 0 ≤ ≤ θ . In the case 1 3
< < θ
the operator H
has the following fixed points: ( )
1 1 ≡ t ϕ , ( ) , 2 1 4 3 0 0 2 − + =
y x t θ ϕ ( )
3 0 0 2 2 1 4 − − = t y x t θ ϕ . Theorem 8. Let 𝑘𝑘 ≥ 2. For the model (6) defined on Cayley tree of order k the following statements hold: а) If k 3 5 0 ≤ ≤ θ , then there exists a unique translation-invariant Gibbs measure; b) If 1 3 5 < < θ
, then there exist three translation-invariant Gibbs measures. In the forth section of Chapter 2, it was proved the existence a phase transition for the model defined on Cayley tree of order 𝑘𝑘 ≥ 2.
Let k = 2 and
1 ≥ n . We defined the operator 𝑉𝑉 2,𝑛𝑛
: (𝑥𝑥, 𝑦𝑦) ∈ ℝ 2 → (𝑥𝑥 ′ , 𝑦𝑦
′ ) ∈ ℝ
2 by
⋅ ⋅ + + ⋅ = ⋅ + + + = + y x n n y y n n x x n θ θ 3 2 1 2 2 ' , 4 3 2 1 2 ' 2 2 1 2 2 39
) 1
( 2 3 2 0 + + ≤ ≤ n n θ
two has a unique (nontrivial) positive fixed point; ii) If 1 ) 1 2 ( 2 3 2 < < + + θ n n , then the Hammerstein operator of order two has exactly three positive fixed points. Consequently, by Proposition 9 the operator H k has the unique positive fixed point ( )
1 1 ≡ t ϕ if 2 3 0 2(2 1)
n θ + ≤ ≤ + . In the case 2 3 1 2(2 1)
n θ + < < + the operator H k has
the following fixed points: ( )
1 1 ≡ t ϕ and , 2 1 2 1 2 ) 3 2 ( ) 1 2 ( 2 1 ) 1 2 ( 2 3 2 ) ( 1 2 2 − ⋅ + + − ⋅ + + ⋅ ⋅ + + = + n t n n n n n t θ θ ϕ
2 1 3 2 3 2(2
1) (2 3) 1 ( )
1 2 . 2(2 1) 2 1 2
n n n t t n n θ ϕ θ + + + ⋅ − + = ⋅ −
⋅ − + ⋅ +
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